Draw the one-edge extension between unreachable vertices

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WilliamsonShmoys.planar_draw_one_edge_extension__p2m_8ebe9775_f6af_4eb7_9344_68ad2c5d7599_a4

by Eyal1990 · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

approximation-algorithmsplanar-graphstreewidth

A drawing of a finite graph can be replaced by a drawing of its extension with the edge between two distinct unreachable vertices.

Preamble

import Definitions.Def_WilliamsonShmoys_PlanarDrawingOn
import Mathlib

set_option autoImplicit false
open WilliamsonShmoys

Formal statement

theorem WilliamsonShmoys.planar_draw_one_edge_extension__p2m_8ebe9775_f6af_4eb7_9344_68ad2c5d7599_a4
    {V : Type} [Fintype V] [LinearOrder V] (N : SimpleGraph V)
    (hV : 3 <= Fintype.card V) (hN : HasPlanarDrawingOn N)
    (u v : V) (huv : u ≠ v) (hUnreachable : ¬ N.Reachable u v) :
    HasPlanarDrawingOn
      (N ⊔ SimpleGraph.fromRel (fun a b => a = u ∧ b = v)) := by sorry

Source

David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).

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